Applications of maximum matching by using bipolar fuzzy incidence graphs

The extension of bipolar fuzzy graph is bipolar fuzzy incidence graph (BFIG) which gives the information regarding the effect of vertices on the edges. In this paper, the concept of matching in bipartite BFIG and also for BFIG is introduced. Some results and theorems of fuzzy graphs are also extended in BFIGs. The number of operations in BFIGs such as augmenting paths, matching principal numbers, relation between these principal numbers and maximum matching principal numbers are being investigated which are helpful in the selection of maximum most allied applicants for the job and also to get the maximum outcome with minimum loss (due to any controversial issues among the employees of a company). Some characteristics of maximum matching principal numbers in BFIG are explained which are helpful for solving the vertex and incidence pair fuzzy maximization problems. Lastly, obtained maximum matching principal numbers by using the matching concept to prove its applicability and effectiveness for the applications in bipartite BFIG and also for the BFIG.


Introduction
A graph is more suitable to explain any kind of information along with the mutual relationship between different types of objects. The relationship between different entities are represented in terms of edges while entities do represent vertices. Zadeh was the first one who introduced the theory of fuzzy sets (FSs), which provides us the grade of membership of an object [1]. This theory opened an energetic area of research in various disciplines in the fields of automata, medical sciences, computer networking, statistics, social sciences and its various subbranches and disciplines, management sciences, engineering and graph theory etc. In this way Zadeh was the one who paved the way for Rosenfeld who introduced the fuzzy graph (FG) theory [2]. Rosenfeld was the one who presented several graph theoretical ideas, for example path, cycle and connectedness. FG is used when there is an inadequacy in the explanation and justification to various objects or entities and it was of great help to researcher. Mordeson put forth the concept regarding FGs and determined basic properties of it [3].
FGs are unable to give the detailed information about the impact of vertices on edges. This shortage in FGs was the basic problem which is covered by fuzzy incidence graphs (FIGs). The concept of FIG was put forth by Dinesh [4]. Different concepts with regard to the connectivity were put forth by Moderson and Mathew [5]. They introduced various structural properties and establish the prevalence of a strong path between any of the node are pair of a FIG. Inter connectivity between index and wiener index with regard to the FIGs was put forth by Fang et al. [6].
Zhang introduced the concept of bipolar fuzzy sets (BFSs) [7]. The membership grade in the extension of FS to BFS is [-1,1]. An element has 0 grade in BFS if it has zero role on the resultant property. In such a way the membership degree of an element would be(0, 1] which will explain its properties to some extent. If membership grade of an element is [−1, 0) which tells that its marginal pleases the implicit counter property. The idea of the symbolization of bipolar fuzzy graphs (BFGs) along with the matrices in FGs, regular and irregular BFGs, hyper BFGs and antipodal BFGs along with their various applications, properties and significance was explained by Akram et al. [8][9][10][11][12][13][14]. Mohanta et al. gave a study of m-polar neutrosophic graph with applications [15]. Xiao et al. gave the study on regular picture fuzzy graph with applications in communication networks [16].
FG gives only positive membership values of vertices and edges whereas FIGs gives the positive membership values of vertices, edges and incidence pairs. BFG are able to give positive and negative membership values of vertices and edges. FGs, FIG and BFGs are unable to give the detailed information about the impact of vertices on edges. This shortage in BFGs was the basic problem which is covered by BFIGs. BFGs are able to give positive and negative membership values of vertices and edges whereas BFIGs are able to give positive and negative membership values of vertices, edges and incidence pairs. The concept of BFIG was put forth by Gong and Hua [17]. There are multiple reasons to introduce the concept of matching in bipartite BFIG and for BFIG. Let us consider an example to understand the concept of BFIG, if nodes reflects distinct companies and edges are the roads which connects these companies, then an BFG will give us the information of traffic between these companies. The company which have more number of employees will have the foremost infrastructure in the company. Hence, if C 1 and C 2 be two companies and C 1 C 2 is a road between these companies, then (C 1 , C 1 C 2 ) will be the incline system from the the company C 1 using the road C 1 C 2 to the company C 2 . Similarly, (C 2 , C 1 C 2 ) will be the incline system from the company C 2 using the road C 1 C 2 to the company C 1 . Both C 1 and C 2 have the impact of 1 on C 1 C 2 in un-weighted graphs. But, the impact of C 1 on C 1 C 2 will be (C 1 , C 1 C 2 ) is 1 whereas (C 2 , C 1 C 2 ) is 0 in a directed graph. This is the main concept of BFIG.
Matching is important area in the graph as well as in the FG theory. It was Shen and Tsai who introduced the concept of optimal graph matching approach for solving the task assignment problem [18]. The concept of matching in FGs was introduced by Ramakrishnan and Vaidyanathan [19]. Later on, Mohan and Gupta further worked and gave the Graph matching algorithm for task assignment problem [20]. Matching numbers in fuzzy graphs are explained by Khalili et al. [21]. Our first objective is to find out maximum matching principal numbers in bipartite BFIG and for BFIG which are helpful to reflect the selection maximum applicants and their maximum working with minimum loss due to some controversial issues. Besides of this, some of the characteristics of the matching as well as bounds in bipartite BFIGs and BFIG have also been discussed. By using related examples, a detailed study has been carried out in the fields of matching number for the BFIGs.
Section 2 gives some preliminary definitions which are helpful to understand the next sections of the article. Section 3 contains some definitions, examples, results and theorems related to the concept of matching in BFIG. Section 4 gives mathematical model for obtaining MMVBFIN and MMBFIN for bipartite BFIG and BFIG. Section 5 contains comparative analysis is discussed for matching in bipartite BFIG and BFIG. Lastly, conclusions and prospects are explained in section 6.

Bipolar fuzzy incidence graph
This segment consists of some basic definitions including FS, BFS, FG, incidence graph (IG), FIG, BFG, BFIG, complete bipolar fuzzy incidence graph (CBFIG), matching, some concepts related to matching in classical theory and some examples. In this article, V, E = V × V and I = V × E represents the set of vertices, set of edges and set of incidence pairs, respectively. Let G = (V, E) be a crisp graph. A set _ M of pairwise non-adjacent edges is known as matching. A matching _ M is known to be perfect matching if it covers all the vertices of the crisp graph G and if a matching _ M covers maximum vertices then it is known as maximum matching. A crisp graph G is said to be nearly perfect matching if only one vertex is unmatched. The number of edges in a maximum matching is known as the matching number and is denoted by að _ MÞ. A track in which edges are alternating in _ M and E À _ M is known as _ M-alternating track and if neither its starting and nor its final vertex is covered by _ M then, it is known as _ M-augmented track.
Definition 2.1: Definition 2.2: [1] Let V be the any nonempty set from the universal set U, a mapping U : V ! [0, 1] is known as fuzzy subset. Definition 2.3: [2] Let v be the fuzzy subset of the set V and E be the fuzzy subset of V × V.

Matching in bipolar fuzzy incidence graph
This segment consists of some definitions like, support of BFIG, degree of vertices, degree of edges and incidence pairs in BFIGs, path, strength, strength of connectedness, matching, matching principal numbers, maximum matching principal numbers, some examples and theorems. Definition 3.1: Let _ G ¼ ðV; E; IÞ be the BFIG, then the support of BFIG is denoted by _ G ¼ ðV * ; E * ; I * Þ and is defined as Definition 3.2: Let _ G ¼ ðV; E; IÞ be the BFIG.
• Two vertices v 0 and v 1 are said to be connected if there exist a path from v 0 to v 1 such that v 0 , • The degree of any incidence pair Similarly, the degree of distinct incidence pairs is given as:

Definition 3.5: The strength of connectedness between
Proof As a matching is taken as the set of triples like h. . ., v i e j v k , . . .i and we must mention the vertices and incidence pair specifically. So, a matching M as presented in Fig 3 and can be written as h If there is a path which connects v i and v j , then this path is a single incidence pair • The matching bipolar fuzzy incidence number of _ M can be described as, • The matching edge bipolar fuzzy incidence number of _ M can be described as, • The matching vertex bipolar fuzzy incidence number of _ M can be described as, • The matching crisp number of _ M can be described as, a C ð _ MÞ ¼ jIð _ MÞj.
We consider a I ð _ MÞ, a V ð _ MÞ and a C ð _ MÞ as matching bipolar fuzzy incidence principal numbers (MBFIPNs) of _ M. • The maximum matching bipolar fuzzy incidence number of _ G can be described as: • The maximum matching edge bipolar fuzzy incidence number of _ G can be described as: • The maximum matching vertex bipolar fuzzy incidence number of _ G can be described as: • The maximum matching crisp number of _ G can be described as: We M. So, we have:   ; v nþ1 g. Now, by using definition 3.11, we have:  Table 2     Step-3 By using third step, we get _ M 2 which is _ M 2 ¼ hðu 1 ; u 1 v 2 Þ; ðu 2 ; u 2 v 1 Þi and its augmenting path is represented by (u 3 , u 3 v 0 ). Now, obtaining again its augmenting path presented in Fig 9; Step-3 By using third step, we get _ M 3 which is _ M 3 ¼ hðu 3 ; u 3 v 0 Þi and its augmenting path is represented by (u 1 , u 1 v 2 ), (u 2 , u 2 v 1 ) which is same as _ M 2 as shown in Fig 10. Hence, by using step-4, we get _ M 1 as our final matching and MMVBFIN = (2.7, −2.1).

MMVBFIN problem in arbitrary BFIG
In this part, we are explaining the process to obtain the MMVBFIN in the arbitrary BFIG. In this process the main points are; Step-1 Arrange the vertices such that v 1 is strongest vertex. v 2 is the vertex which is connected with v 1 and weaker than v 1 . Step-2 Consider an incidence pair, except (v 1 , v 1 v 2 ) connected with v 2 which is our first matching. If, no such incidence pair is found then, start from (v 1 , v 1 v 2 ).
Step-3 Obtain the strong vertex augmenting path from _ M 1 and continue this process untill there is no augmenting path.
Step-4 Choose the maximum vertex matching and obtain MMVBFIN. Application Let _ G be the BFIG as shown in Fig 11. Step-1 By using step one, v 1 is strongest vertex. v 2 is the vertex which is connected with v 1 and weaker than v 1 . v 3 is weaker then v 2 and so on.
Step-2 The incidence pair _ M 1 ¼ ðv 2 ; v 2 v 3 Þ is our first matching and the augmenting path Step Þi is second matching and the augmenting path is obtained MMBFIN = (1, −0.6) and got the same matching _ M 1 ¼ hðu 1 ; u 1 v 3 Þ; ðu 3 ; u 3 v 2 Þ; ðu 2 ; u 2 v 0 Þi for BBFIG by using the incidence pairs. So, the result is better by using the incidence pairs as by using the vertices we have more chances of controversial issues i.e., MMVBFIN ¼ 2:1 2:7 * 100 ¼ 77:78 and by using the incidence pairs, we have MMBFIN ¼ 0:6 1 * 100 ¼ 60. Now, there are six employees in a company. We give them membership values according to their individual performance. There edge values are defined as their work performance with other member as a group. The positive membership value of incidence pair value defines the working efficiency of two employees in a company and negative membership value defines their loss possibility in the working due to controversial issues among the employees as a group. In Both matchings either by using vertex or incidence pairs are same but incidence pairs are representing the influence of on vertex to other. So, the incidence graphs are more better as by using the incidence pairs we can see that which employee have greater influence on other or which group have better efficiency level.

Conclusion
Graph theory is very needful for presenting the data of real life problems. In this article, we enhanced the theory of BFIGs. The matching concept becomes very useful when it is discussed by using BFIGs because it also includes the controversial issues or chances of loss among the employees in a company. After introducing the concept of matching in BFIGs, its related propositions, results and theorems with some examples are presented. Matching numbers are obtained to improve the working quality of the employees in a company. Finally, a decision making graph of a company is presented to reflect the working of the members and achieving maximum results by minimizing chances of loss. Our goal is to enhance this research to soft FIGs, q-rung FIGs with more theorems and applications in forthcoming articles.